Math101 · Independent Ontario learning library
Math101Mean
The arithmetic mean is a balance-point average found by sharing a total equally across observations.
The arithmetic mean equals the total of all observations divided by the number of observations.
The equal-share idea
For data $x_1,x_2,\ldots,x_n$, the mean is
Imagine redistributing the total so every observation receives the same amount. That common amount is the mean. The total is preserved: $n\bar{x}=\sum x_i$.
A basic calculation
Keep the numerator and count visible. Most errors come from an incomplete total or wrong number of observations.
Finding a missing value
If the mean and count are known, recover the total first. Five values with mean $12$ must total $60$. If four known values total $47$, the missing value is $60-47=13$.
This approach is more reliable than creating an equation around a memorized average formula without interpreting the total.
Context and units
The mean has the same unit as the data. A mean time is measured in seconds or minutes, not squared units. Interpret it with population, measurement method, and spread; two groups can share the same mean but differ greatly in consistency.
Common mistakes
Dividing by the wrong count. Count observations, including repeated values.
Averaging averages without weights. Group sizes may differ.
Assuming the mean must be observed. It can lie between data values.
Calling it “typical” without checking shape or outliers. Compare with median and spread.
Quick self-check
- Did I include every observation exactly once?
- Is the divisor the number of observations or total frequency?
- Are weights represented correctly?
- Could an outlier make median a better summary?
